1 Introduction
Population protocols were introduced by Angluin et al. in [DBLP:conf/podc/AngluinADFP04, DBLP:journals/dc/AngluinADFP06] to study the computational power of networks of small resourcelimited mobile agents. In this model, each agent has a state in a finite set of states. When agents interact, their states are updated accordingly to a finite interaction table. This table corresponds intuitively to a conservative Petri net (the number of agents is preserved by each transition) where each line of the interaction table is matched by a transition of a Petri net.
Population protocols decide properties by stable consensus. Intuitively, from any initial configuration extended with some some extra agents called leaders, and under some fairness conditions, in any infinite execution either all agents eventually accept or all agents eventually reject. The population protocol is said to be wellspecified if the resulting consensus does not depend on the computation, but only on the initial configuration. Deciding if a population protocols is wellspecified was proved to be decidable in [DBLP:conf/concur/EsparzaGLM15, DBLP:journals/acta/EsparzaGLM17] by observing that the wellspecified property is equivalent to the reachability problem for Petri nets up to elementary reductions. Since this last problem was recently proved to be Ackermanniancomplete [DBLP:journals/corr/abs210412695, DBLP:journals/corr/abs210413866], it means that deciding the wellspecified property of a population protocol is also Ackermanniancomplete. In particular, population protocols maybe intrinsically very complicated.
In this paper, population protocols are assumed to be wellspecified and the set of initial configurations accepted by a population protocol is called the decided predicate. In [DBLP:journals/dc/AngluinAER07], Angluin et al. have shown that predicates decidable by population protocols are exactly the predicates definable in Presburger arithmetic. It follows that the state complexity of a Presburger predicate defined as the minimal number of states of a population protocols deciding it is well defined. In [DBLP:conf/stacs/BlondinEGHJ20], by revisiting the construction of population protocols deciding Presburger predicates, some improvement on state complexity upperbounds was derived.
Computing state complexities is a difficult problem. It follows that focusing on the state complexity of simple Presburger predicates is a natural question. The simplest non trivial Presburger predicates are clearly the counting predicates that corresponds to the set of configurations such that the number of agents in a given state is larger than or equal to . In [DBLP:conf/stacs/BlondinEJ18], it was exhibited an infinite set of natural numbers for which the associated counting predicates can be decided by population protocols with states. Moreover, very recently in [DBLP:conf/podc/CzernerE21] the state complexity of population protocols deciding the counting predicate was proved to be at least where is some Ackermannian function. It follows that there is a gap between the known upper and lower bounds for the state complexity of counting predicates.
In this paper, we close this gap by proving that any population protocol deciding the counting predicate associated to a natural number requires at least states. Since our proof techniques allow some generalizations of the model of population protocols, we present our result for population protocols that allow leaders, agent destruction and creation. We call the classical population protocols the conservative ones. Compared to conservative population protocols, the number of reachable configurations from an initial configuration is no longer finite. Due to this observation, the notion of fair executions and wellspecified population protocols are (briefly) revisited in this paper. It worth mentioning that our definitions match the classical ones for conservative population protocols.
In Section 2 we recall some basic definitions and results about Petri nets. In Section 3 we introduce the notions of stabilized, stable, and bottom configurations. The two first definitions are central for defining the semantics of (non conservative) population protocols and the last one is used for proving our lower bound. In Section 4 we introduce our generalized model of population protocols that allow leaders, agent destruction and creation. In Section 5 we provide bridges between our definitions and the classical definitions of population protocols. Section 6 contains the central technical lemma. It is a lemma about Petri nets that intuitively shows that from any initial configuration we can reach with short execution kind of bottom configurations. Section 7 recalls the model of Petri net with states, and provides a result on small cycles satisfying some properties. This last result is obtained by introducing a linear system and by applying Pottier’s techniques [Pottier:1991:MSL:647192.720494] in order to obtain small solutions for that linear system. Results of the previous sections are combined in Section 8 to obtain the state complexity lower bound . Some related open problems and future work are presented in Section 9.
2 Petri Nets
We first recall some basic definitions and results about Petri nets.
We assume that an infinite countable set of elements called counters is fixed. A configuration is a mapping such that the set is finite. We denote by the zero configuration, i.e. the unique one satisfying . Given a counter , we denote by the unit configuration defined by and for every counter . We associate with a configuration the numbers and . The sum of two configurations is the configuration defined pointwise by for every counter . The partialorder over the configurations is also defined pointwise by if for every counter .
A transition is a pair of configurations, where is called the precondition, and the postcondition. We introduce and . We associate with a transition the binary relation over the configurations defined by if there exists a configuration such that and . Intuitively a transition first removes from the precondition to get the intermediate configuration and then add the postcondition to get . We associate with a word of transitions the binary relation over the configurations defined by if there exists a sequence of configurations such that:
A Petri net is a finite set of transitions. We introduce , and the set of words of transitions in . We also define if is a non empty Petri net and zero otherwise. We denote by the binary relation over the configurations defined by if there exists a word such that . This relation is called the reachability relation of . When we say that is reachable from .
A transition is said to be conservative if . A Petri net is said to be conservative if every transition in is conservative. Notice that every configuration that is reachable from satisfies for any conservative Petri net . In particular the set of configurations that are reachable from any configuration is finite for every conservative Petri net .
We say that a configuration is coverable from a configuration if there exists a configuration that is reachable from . Notice that a configuration is coverable from a configuration if, and only if, there exists a word such that for some configuration . In that case, the minimal length of such a word can be bounded using Rackoff’s techniques with respect to , , and . We now recall those techniques.
Given a set of counters, we define several restrictions related to as follows. Given a configuration , we denote by the configuration defined by if , and zero otherwise. Given a transition , we define the transition as the pair . Given a Petri net , we introduce the Petri net . Given a word of transitions, we introduce the word . Notice that implies . The converse property is true in some cases as shown by the following lemma. Assume that for some configurations , some word of transitions in a Petri net , and some set of counters. If for every counter then there exists a configuration such that , , and for every counter .
Proof.
Simple induction on . ∎
We are now ready to recall a classical result used in [DBLP:journals/tcs/Rackoff78] to prove that the coverability problem is decidable in exponential space. [[DBLP:journals/tcs/Rackoff78]] If a configuration is coverable from a configuration , then there exists with a length bounded by where , and a configuration such that .
Proof.
This is a classical result obtained by Rackoff in [DBLP:journals/tcs/Rackoff78] by induction on . A complete proof with our notations is recalled in appendix. ∎
3 Stabilized, Stabilizable, and Bottom Configurations
We introduce the notion of stabilized, stabilizable, and bottom configurations. We assume that is a Petri net.
A configuration is said to be stabilized for a set of counters if for every configuration that is reachable from . The following lemma shows that stabilized configurations are characterized by the values of the “small counters”. Let be a stabilized configuration, let be a positive integer satisfying where , and let . Every configuration such that is stabilized.
Proof.
Let us consider a configuration such that . There exists a configuration such that . Assume by contradiction that is not stabilized. It follows that there exists a configuration that is reachable from and such that for some counter . Since and is stabilized, we deduce that . In particular since . It follows that the unit configuration is coverable from since . Lemma 2 shows there exists a word of length bounded by and a configuration such that . It follows that . From this relation and , we deduce that . Lemma 2 shows that there exists a configuration such that and . Since , we deduce that . It follows that . As , it follows that is not stabilized and we get a contradiction. We have proved the lemma. ∎
A similar result was provided in [DBLP:conf/podc/CzernerE21] in the context of conservative Petri nets.
A configuration is said to be stabilizable if every configuration that is reachable from , there exists a stabilized configuration that is reachable from . Clearly for any configuration there exists a finite set such that is stabilizable just by considering for the set . The following lemma shows that in fact there exists a unique minimal for the inclusion set such that is stabilizable. We denote by this set. If a configuration is stabilizable and stabilizable then it is stabilizable.
Proof.
Let be a configuration reachable from . Since is stabilizable, there exists a stabilized configuration that is reachable from . Since is stabilizable, it follows that there exists a stabilized configuration that is reachable from . Let us prove that is stabilized with . To do so, let be a configuration reachable from . Since is stabilized, we get . Moreover, since is reachable from we deduce that . It follows that . We have proved that is stabilized. It follows that is stabilizable. ∎
The component of a configuration is the set of configurations such that . A configuration is said to be bottom if its component is finite and every configuration such that satisfies .
4 Population Protocols
In this section, we introduce our model of population protocols and some related notions.
A population preprotocol is a tuple where is a Petri net, is a configuration called the configuration of leaders, are two finite sets of counters. When , the preprotocol is said to be leaderless. When is a conservative Petri net, the preprotocol is said to be conservative. A preprotocol is said to be correctlyspecified if for every configuration such that the zero configuration is not reachable from and the set is included in or disjoint from . A correctlyspecified population preprotocol is simply called a population protocol. Given a population protocol , we introduce the set of configurations such that and such that . The set is called the predicate decided by . We also introduce .
A counting predicate where is a positive natural number is a set of configurations of the form for some counter . It is proved in [DBLP:conf/stacs/BlondinEJ18] that there exists an infinite set of positive numbers such that for every , there exists a conservative population protocol deciding a counting predicate such that . In this paper, we show that this result is almost optimal by proving the following theorem. For every population protocol deciding a counting predicate, we have the following bound where :
For leaderless population protocols, Example 4 shows that there exist conservative leaderless population protocols deciding counting predicates with when is a power of two. In fact, following [DBLP:conf/stacs/BlondinEJ18] such a construction can be extended in such a way for every natural number , there exists a conservative leaderless population protocol deciding counting predicate with . Proving that for every leaderless conservative population protocol deciding the counting predicate is still a problem left open in this paper.
[[DBLP:conf/stacs/BlondinEJ18]] The counting predicates can be decided by conservative leaderless population protocols as follows. We consider distinct counters denoted as , and we let be the set of those counters. We introduce the Petri net where:
Just observe that is a conservative leaderless population protocol deciding .
5 Bridges to Classical Definitions
In this section, we provide bridges between our definitions of population protocols and the classical ones that are only defined for conservative population protocols. This section can be freely skipped since definitions and results provided in this section are not used anywhere else in the paper.
Let be a Petri net. A execution from an initial configuration is an infinite sequence of configurations such that for every either , or there exists a transition such that . The set of counters limitused by a execution is the set of counters such that for infinitely many .
Following [DBLP:conf/podc/CzernerE21], a execution is said to be fair if for every configuration we have:
Let us recall this classical result about fair executions. Let be a conservative Petri net. For every fair execution , there exists such that is bottom.
Proof.
Since is conservative, the set of configurations reachable from is finite. It follows that there exists a configuration such that for infinitely many . Observe that the component of is finite since is conservative. Now, let be a configuration reachable from . Observe that for every such that , we have . Since the execution is fair, we derive that there exists such that . Since for infinitely many , there exists such that . From , we derive . We have proved that is bottom. ∎
Given a configuration , we introduce the set as the union of the sets of counters limitused by fair executions from . Let be a conservative Petri net. We have for every configuration .
Proof.
Let .
Let us first consider a counter . It follows that there exists a fair execution such that for infinitely many . Lemma 5 shows that there exists such that is bottom. It follows that is bottom for any . In particular, we can assume that . Since is stabilizable, it follows that is stabilizable as well. In particular, there exists a stabilized configuration that is reachable from . As is bottom, the configuration is reachable from . It follows that is stabilized. As we deduce that .
Conversely, let us consider a counter . Let be the set of counters except . By minimality of , Lemma 3 shows that is not stabilizable. It follows that there exists a configuration that is reachable from , such that every configuration that is reachable from is not stabilized. As is conservative, the set of configurations reachable from is finite. In particular there exists a bottom configuration that is reachable from . By replacing by this configuration, we can assume without loss of generality that is bottom. Moreover, as is not stabilizable, there exists a configuration reachable from that uses . Once again, by replacing by this configuration, we can assume that . Now, observe that there exists a fair execution from such that all the configurations of the component of are repeated infinitely often. Since , we deduce that the set of counters limitused by this execution contains . Hence . ∎
Following [DBLP:conf/concur/EsparzaGLM15], a conservative population preprotocol is said to be wellspecified if for every configuration such that the set is included in or disjoint from . The set of configurations such that and , is called the predicate accepted.
The following lemma provides the bridges. A conservative population preprotocol is wellspecified if and only if it is correctlyspecified. Moreover, in that case the predicate decided and the predicate accepted coincide.
Proof.
This is a direct corollary of Lemma 5. ∎
6 Small Bottom Configurations
In this section we prove the following theorem that intuitively provides a way to reach with short words kind of bottom configurations with small size (small and short meaning doublyexponential in that context). Other results proved in this section are only used for proving this theorem and are no longer used in the sequel. Let be a Petri net, let , let be a configuration, and let . There exist two words , a set of counters , and two configurations such that:

.

.

for every counter .

is bottom.

The cardinal of the component of is bounded by .

.
The proof of the previous theorem is obtained by iterating the following lemma in order to obtain an increasing sequence of sets . Let be a Petri net, let be a configuration, let be a set of counters included in such that is bottom, let be the cardinal of the component of , and let .
There exist a word such that , and a configuration such that and such that:

either and for every counter ,

or there exists a set that strictly contains such that is bottom and the cardinal of the component of satisfies:
Proof.
Let us introduce the sequence of natural numbers satisfying and satisfying for every . Observe that . Moreover, for every . We deduce by induction that .
Let . We are going to build by induction on a sequence of configurations, a sequence of words in , and a sequence of distinct counters in such that for every we have:

.

.

for every .
So, let us assume that , , and are built for some . Since are distinct elements in , it follows that .
Let us first assume that . In that case, we have for every . As is a bottom configuration and and since the cardinal of the component of is bounded by , we deduce that there exists a word such that and . Since , Lemma 2 shows that for some configuration such that and such that for every we have by definition of . Let us introduce and notice that and we have proved that the lemma holds (first case).
So we can assume that . Let us introduce the set . Since , the set is non empty.
Assume first that for every configuration such that we have for every . In that case let . It follows that the cardinal of the set of configurations such that is bounded by . Hence, there exists a configuration that is bottom and a word such that and such that . Notice that the cardinal of the component of is bounded by . As for every and , Lemma 2 shows that there exists a configuration such that , and . Let us consider the word . Notice that and we have proved that the lemma holds (second case).
Finally, assume that there exists a configuration such that for some word and such that for some counter . We assume that is minimal. Observe that every intermediate configuration such that with and satisfies for every by minimality of . We deduce that there exists a word such that and such that . As for every and , Lemma 2 shows that there exists a configuration such that and . In particular . By minimality of , we get . Now, observe that for every we have by definition of . We have extended our sequence in such a way , , and are fulfilled.
We have proved the lemma. ∎
Now, let us prove Theorem 6. Observe that if the theorem is trivial. So, we can assume that . Let , , and . Notice that is bottom and the cardinal of the component of contains elements. We build by induction on a sequence of subsets of , a sequence of configurations, a sequence of words in such that for every :

.

is bottom.

The cardinal of the component of is equal to .

.

.
Assume the sequence built for some . Lemma 6 on the configuration and the set shows that there exist a word such that , and a configuration such that , such that:

either and for every ,

or there exists such that such that is bottom and the cardinal of its component satisfies:
Observe that in the second case we have extended the sequences. In the first case, let , , , , and . Since are subsets of , we deduce that . Let us introduce and and let us prove by induction on that we have and with the convention .
The rank is immediate. Assume the rank proved. We have:
Since , we deduce that . The induction is proved.
It follows that , , and . Since , we deduce that . In particular . We have proved Theorem 6.
7 Petri Nets with States
A Petri net with states is a triple where is a non empty finite set of elements called states, is a Petri net, and
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